Nuclear potential energy

A harmonic approximation has usually been used for the description of the nuclear potential energy,... 

The terms on the right-hand side of eq. (11.41) denote the kinetic energy, the electron-nuclear potential energy, the Coulomb (J) and exchange (K) terms respectively. Together J and K describe an effective electron-electron interaction. The prime on the summation in the expression for K exchange term indicates summing only over pairs of electrons of the same spin. The Hartree-Fock equations (11.40) are solved iteratively since the Fock operator / itself depends on the orbitals iff,. 

Using the relationship between force and potential energy discussed earlier, we can represent the nuclear force in terms of a simple plot of the nuclear potential energy as a function of distance to the center (Fig. 5.1). Since low-energy particles... 

Ia) is included in the electronic Hamiltonian since, as we shall see, its most important effects arise from interactions involving electronic motions. The interactions which arise from electron spin, 30(5 ), will be derived later from relativistic quantum mechanics for the moment electron spin is introduced in a purely phenomenological manner. The electron-electron and electron-nuclear potential energies are included in equation (2.36) and the purely nuclear electrostatic repulsion is in equation (2.37). The double prime superscripts have been dropped for the sake of simplicity. We remind ourselves that // in equation (2.37) is the reduced nuclear mass, M M2/(M + M2). 

It is of considerable importance to note that the density-potential relationship (3) of the TF theory follows from a variational principle for the total energy. To see this, we note first that the classical electrostatic potential energy U consists of the sum of two terms in an atomic ion, the electron-nuclear potential energy Ken and the electron-electron potential energy Kee. We can write... 

Thus, we have in this model a nuclear potential energy Vx(r) given by... 

There are two curves because the nuclear-nuclear potential energy for the tetrahedral and octahedral nuclear frameworks differ by different values of constant c in equation (91)... 

Figure 5 Quantity 2Vee+ Ven with Vtn the electron-nuclear potential energy against total kinetic energy T for light molecules. Energies are in Hartree units...

Regularities in Nuclear-Nuclear Potential Energy.—Mucci and March first noted, in their study of regularities in the nuclear-nuclear potential energy Knn for tetrahedral and octahedral molecules, that for the cential field model of Section 9 one could combine equations (91) and (92) to obtain for Van at equilibrium... 

Figure 6 Nuclear potential energy Vnn for tetrahedral and octahedral molecules as given by equation (99)...

Figure 7 Nuclear-nuclear potential energy Vnn against total number of electrons N for tetrahedral and octahedral molecules. In contrast to Figure 6, empirical data for bond lengths are used to construct this figure...

But March and Parr12 have gone further, and interpreted the Teller theorem as showing that, in the limit of a large number N of electrons, the nuclear-nuclear potential energy is a smaller term in the number of electrons than the electronic energy terms. This proposal is supported by the fact that the curve of Mucci and March shown in Figure 7 can be approximately represented by... 

March and Parr12 also consider the chemical potential in the same limit. They argue that the meaning of = 0 in the Euler equation of the density description is that Np in this equation is a smaller-order term in the number of electrons than the other energy components. Thus gross features, of the kind exhibited in the energy relations (96)—(98), can be treated but the chemical potential and the nuclear-nuclear potential energy, require special care. 

As in the case of atoms, one can determine the individual components of the energy from this expression. Thus, using Feynman s theorem one can obtain the electron-nuclear potential energy as... 

The symbol V will be used to denote the complete potential energy operator, the sum of the electron-nuclear V , electron-electron and nuclear-nuclear potential energy operators,... 

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